2 0 Ja n 20 10 On the inference of large phylogenies with long branches : How long is too long ? ∗

The accurate reconstruction of phylogenies from short molecular sequences is an important problem in computational biology. Recent work has highlighted deep connections between sequence-length requirements for highprobability phylogeny reconstruction and the related problem of the estimation of ancestral sequences. In [Daskalakis et al.’09], building on the work of [Mossel’04], a tight sequence-length requirement was obtained for the simple CFN model of substitution, that is, the case of a two-state symmetric rate matrix Q. In particular the required sequence length for high-probability reconstruction was shown to undergo a sharp transition (from O(log n) to poly(n), where n is the number of leaves) at the “critical” branch length gML(Q) (if it exists) of the ancestral reconstruction problem defined roughly as follows: below gML(Q) the sequence at the root can be accurately estimated from sequences at the leaves on deep trees, whereas above gML(Q) information decays exponentially quickly down the tree. Here we consider a more general evolutionary model, the GTR model, where the q× q rate matrix Q is reversible with q ≥ 2. For this model, recent results of [Roch’09] show that the tree can be accurately reconstructed with sequences of length O(log(n)) when the branch lengths are below gLin(Q), known as the Kesten-Stigum (KS) bound, up to which ancestral sequences can be accurately estimated using simple linear estimators. Although for the CFN model gML(Q) = gLin(Q) (in other words, linear ancestral estimators are in some sense best possible), it is known that for the more general GTR models one has gML(Q) ≥ gLin(Q) with a strict inequality in many cases. Here, we show that this phenomenon also holds for phylogenetic reconstruction by exhibiting a family of symmetric models Q and a phylogenetic reconstruction algorithm which recovers the tree from O(log n)-length sequences for some branch lengths in the range (gLin(Q), gML(Q)). Second we prove that phylogenetic reconstruction under GTR models requires a polynomial sequence-length for branch lengths above gML(Q).